Arbitrarily Large Polya groups of compositum of simplest fields and some bi-quadratic Polya fields

The Pólya group Formula: see text of an algebraic number field Formula: see text is the subgroup of the ideal class group Formula: see text generated by the ideal classes of the products of prime ideals of the same norm. If Formula: see text is trivial, then the number field Formula: see text is said to be a Pólya field. In the first part of this article, we extend a recent result of ours and prove the existence of infinitely many number fields having Pólya groups as large as possible and those number fields being the compositum of a simplest cubic field and a quintic field. Using a result of Zylinski, we also prove that those fields are of index Formula: see text. Next, we produce three new families Formula: see text, Formula: see text and Formula: see text of totally real bi-quadratic Pólya fields Formula: see text involving prime numbers Formula: see text and Formula: see text that satisfy certain quadratic residue conditions among themselves. It is worthwhile to note that in each of these fields, exactly five primes ramify in Formula: see text and this is the maximum possible number of ramified primes in a bi-quadratic Pólya field over Formula: see text.